derivative of ln(ln(ln(x)))

Steps to Solve

1. Identify the function and the outermost derivative We start with the function $y = \ln(\ln(\ln(x)))$. The outermost function is the natural logarithm, so we will use the derivative of the natural logarithm, which is $\frac{1}{u}$, where $u$ is the argument of the logarithm.

2. Apply the chain rule to the outermost function Taking the derivative of $y$ with respect to $x$, we have: $$ \frac{dy}{dx} = \frac{1}{\ln(\ln(x))} \cdot \frac{d}{dx}[\ln(\ln(x))] $$

3. Identify the next inside function and its derivative Now we need to find the derivative of $\ln(\ln(x))$. Again, we apply the chain rule. We denote this new variable as $v = \ln(x)$, so we have: $$ \frac{d}{dx}[\ln(\ln(x))] = \frac{1}{\ln(x)} \cdot \frac{d}{dx}[\ln(x)] $$

4. Calculate the derivative of the innermost function The derivative of $\ln(x)$ is $\frac{1}{x}$. Substituting this into the equation, we find: $$ \frac{d}{dx}[\ln(\ln(x))] = \frac{1}{\ln(x)} \cdot \frac{1}{x} $$

5. Combine everything together Now we substitute back into the equation we found in step 2: $$ \frac{dy}{dx} = \frac{1}{\ln(\ln(x))} \cdot \left(\frac{1}{\ln(x)} \cdot \frac{1}{x}\right) $$

6. Write the final derivative expression This leads us to the final derivative: $$ \frac{dy}{dx} = \frac{1}{x \ln(x) \ln(\ln(x))} $$